Convective planetary boundary layer

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The convective planetary boundary layer (CPBL), also known as the daytime planetary boundary layer (or simply convective boundary layer, CBL, when in context), is the part of the lower troposphere most directly affected by solar heating of the Earth's surface. [1]

Contents

This layer extends from the Earth's surface to a capping inversion that typically locates at a height of 1–2 km by midafternoon over land. Below the capping inversion (10–60% of CBL depth, also called entrainment zone in the daytime), CBL is divided into two sub-layers: mixed layer (35–80% of CBL depth) and surface layer (5–10% of CBL depth). The mixed layer, the major part of CBL, has a nearly constant distribution of quantities such as potential temperature, wind speed, moisture and pollutant concentration because of strong buoyancy generated convective turbulent mixing.

Parameterization of turbulent transport is used to simulate the vertical profiles and temporal variation of quantities of interest, because of the randomness and the unknown physics of turbulence. However, turbulence in the mixed layer is not completely random, but is often organized into identifiable structures such as thermals and plumes in the CBL. [2] Simulation of these large eddies is quite different from simulation of smaller eddies generated by local shears in the surface layer. Non-local property of the large eddies should be accounted for in the parameterization.

Mean Characteristics

The mean characteristics of the three layers of the CBL are as follows.

Vertical profiles of mean variables in convective boundary layer. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 13 Vertical profile of mean variables in convective boundary layer.pdf
Vertical profiles of mean variables in convective boundary layer. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 13

(1) The Surface layer is a very shallow region close to the ground (bottom 5–10% of CBL). It is characterized by a superadiabatic lapse rate, moisture decrease with height and strong wind shear. [2] Almost all wind shear and all the potential temperature gradient in the CBL are confined in the surface layer.

(2) The Mixed layer composing the middle 35–80% of CBL [2] is characterized by conserved variables such as potential temperature, wind speed and moisture.

(3) The Entrainment zone (or Capping inversion) can be quite thick, averaging about 40% of the depth of the CBL. It is the region of statically stable air at the top of the mixed layer, where there is entrainment of free atmosphere air downward and overshooting thermals upward. [2] Potential temperature and wind speed have a sharp increase across the capping inversion, while moisture has a sharp decrease.

Evolution

CBL depth has a strong diurnal cycle with a four-phase process growth: [3]

Evolution of convective boundary layer. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 11 Evolution of convective boundary layer.pdf
Evolution of convective boundary layer. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 11

(1) Formation of a shallow mixed layer: During the early morning the mixed layer is shallow and its depth increases slowly because of the strong nocturnal stable inversion capping. [4]

(2) Rapid growth: By the late morning, the cool nocturnal air has been warmed to a temperature near that of the residual layer, so the thermals penetrate rapidly upward during the second phase, allowing the top of the mixed layer to rise at rates of up to 1 km per 15 minutes. [4]

(3) Deep mixed layer of nearly constant thickness: When the thermals reach the capping inversion at the top of the residual layer, they meet resistance to vertical motion and the mixed layer growth rate rapidly decreases. During this third phase, which spans over most of the afternoon, the mixed layer depth is relatively constant. The lapse rate of temperature in the CBL is 1°/100m. [4]

(4) Decay: Turbulence generated by buoyancy which drives the mixing decays after sunset and CBL collapses as well.

Turbulence in the CBL

In the atmospheric boundary layer, wind shear is responsible for the horizontal transport of heat, momentum, moisture and pollutants, while buoyancy dominates for the vertical mixing. Turbulences are generated by buoyancy and wind shear. If the buoyancy dominates over shear production, the boundary layer flow is in free convection. When shear generated turbulence is stronger than those generated by buoyancy, the flow is in forced convection.

Normalized turbulent kinetic energy generated by buoyancy and shear by surface buoyancy. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 155 Buoyancy shear.pdf
Normalized turbulent kinetic energy generated by buoyancy and shear by surface buoyancy. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 155

In the surface layer, shear production always exceeds buoyancy production because of strong shear generated by surface drag. In the mixed layer, buoyancy generated by heating from the ground surface is the major driver of convective turbulence. [5] Radiative cooling from the cloud tops is also an effective driver of convection. The buoyancy generated turbulence peaks in the afternoon, hence the boundary layer flow is in free convection during most of the afternoon.

The up and downdrafts of boundary layer convection is the primary way in which the atmosphere moves heat, momentum, moisture, and pollutants between the Earth's surface and the atmosphere. Thus, boundary layer convection is important in the global climate modeling, numerical weather prediction, air-quality modeling and the dynamics of numerous mesoscale phenomena.

Mathematical simulation

Conservation equation

To quantitatively describe the variation of quantities in the CBL, we need to solve the conservation equations. The simplified form of the conservation equation for a passive scalar in typical CBL is

where is the mean of quantity , which could be water vapor mixing ratio , potential temperature , eastward-moving and northward-moving wind speed. is the vertical turbulent flux of .

We made several approximations to get the above simplified equation: ignore the body source, Bousinesq approximation, horizontal homogeneity and no subsidence. Bousinesq approximation is to ignore the density change due to pressure perturbation and keep the density change due to temperature change. This is a fairly good approximation in the CBL. The latter two approximations are not always effective in the real CBL. But this is acceptable in theoretical research. Observations show that turbulent mixing accounts for 50% of the total variation of potential temperature in a typical CBL.

However, due to the randomness of turbulences and our lack of knowledge about the exact physical behavior of it, parameterization of turbulent transport is needed in model simulation. Unlike shear dominated turbulence in the surface layer, large eddies associated with rise of warm air parcels that transport heat from hot to cold, regardless of the local gradient of the background environment exit in the mixed layer. Hence the non-local counter-gradient transport should be properly represented in the model simulation.

Several approaches are generally followed in numerical models to obtain the vertical profiles and temporal variations of quantities in CBL. Full mixing scheme for the whole CBL, local scheme for the shear dominated regions, non-local scheme and top-down and bottom up diffusion scheme for the buoyancy dominated mixed layer. In the full mixing scheme, all quantities are assumed to be uniformly distributed and the turbulent fluxes are assumed to be linearly related to height, with a jump at the top. In the local scheme, the turbulent flux is scaled by the local gradient of the quantity. In non-local scheme, the turbulence fluxes are related to known quantities at any number of grid points elsewhere in the vertical. [6] In top-down and bottom-up diffusion, the vertical profile is determined by diffusion from the two directions and the turbulent fluxes in sub-grid scale are derived from known quantities or their vertical derivatives at the same grid point.

Full mixing scheme

Full mixing is the simplest representation of CBL in some global models. Fluxes within this layer are assumed to decrease linearly with height, and the mean variables keep their vertical profile at each simulation time step. [7] All mean variables are uniformly distributed throughout the whole CBL and have a jump at the top of CBL. This simple model has been used in meteorology for a long time, and continues to be a popular approach in some global course resolution models.

Local closure

The local closure K-theory is a simple and effective scheme for shear dominated turbulent transport in the surface layer. K-theory assumes that mixing for heat, water vapor and pollutant concentration occurs only between adjacent layers of the CBL, and that the magnitude of mixing is determined by the eddy diffusion coefficient and local gradients of corresponding scalars . [8]

Where is an "eddy diffusion coefficient" for , which is typically taken as a function of a length scale and local vertical gradients of . For neutral condition, is parameterized using Mixing-Length Theory.

If a turbulent eddy moves a parcel of air upward by amount during which there is no mixing nor other changes in the value of within the parcel, then we define by

where is the von Karman constant empirically derived (0.35 or 0.4).

Mixing-length theory has its own limitation. The theory only applies to statically neutral condition. [9] It biases for statically stable and unstable conditions.

Mixing-length theory fails when the wind speed is uniformly distributed, people use knowledge of turbulent kinetic energy (TKE) to improve parameterization of eddy diffusion coefficient to account for large eddy transport in typical CBL. TKE gives us a measure of the intensity and effectiveness of turbulence and it could be measured accurately.

where is the dimensionless stability function, and is the TKE. The diagnostic equations used to obtain parameters and differ in different TKE closures.

Non-local closure

In buoyancy dominated regions, K-theory fails since it always yield unrealistic zero flux in a uniform environment. The non-local characteristics of large buoyancy eddies are accounted for by adding a non-local correction to the local scheme. The flux of any scalar can be described with [10]

where is a correction to the local gradient to represent the counter gradient flux transport of[ clarification needed ] large-scale eddies. This term is small in stable conditions, and is therefore neglected in such conditions. In unstable conditions, however, most transport is done by turbulent eddies with sizes on the order of the depth of the boundary layer. [10] In such cases,

where is the corresponding surface flux for a scalar , and is a coefficient of proportionality. is the mixed layer velocity scale defined from surface friction velocity and wind profile function at the surface layer top.

The eddy diffusivity for momentum is defined as

where is the von Karman constant, is the height above the ground, is the height of the boundary layer.

Compared to full mixing scheme, the non-local scheme significantly improves simulations of the vertical distributions for NO2 and O3, as evaluated in a study done in summer using aircraft measurements. It also reduces model biases at surface over the U.S. by 2-5 ppb for peak O3 (O3 concentration is 40-60ppb) in the afternoon, as evaluated using ground observations. [7]

Top-down and Bottom-up diffusion

The entrainment fluxes of quantities are not treated in the non-local scheme. In the top-down and bottom-up scheme, both the surface fluxes and the entrainment fluxes are represented. The mean scalar fluxes are the sum of the two fluxes [11]

Where is the height of mixed layer. and are the scalar flux at the top and bottom of CBL and scale as

Modeled vertical profile and turbulent flux of virtual potential temperature. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 457 and Wyngaard and Brost, 1983 Model turbulent.tiff
Modeled vertical profile and turbulent flux of virtual potential temperature. Adapted from Stull 1988 An Introduction to Boundary Layer Meteorology page 457 and Wyngaard and Brost, 1983

Where and are

is the convective velocity scale . is the dimensionless gradient for the bottom up direction, a function of . is the dimensionless gradient for top-down. The vertical profile of and are provided in the Wyngaard et al., 1983 [11]

See also

Related Research Articles

In thermal fluid dynamics, the Nusselt number is the ratio of convective to conductive heat transfer at a boundary in a fluid. Convection includes both advection and diffusion (conduction). The conductive component is measured under the same conditions as the convective but for a hypothetically motionless fluid. It is a dimensionless number, closely related to the fluid's Rayleigh number.

In fluid mechanics, the Grashof number is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

<span class="mw-page-title-main">Boundary layer</span> Layer of fluid in the immediate vicinity of a bounding surface

In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition. The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.

The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow shear term:

<span class="mw-page-title-main">Surface layer</span> Layer of a turbulent fluid affected by interaction with a surface

The surface layer is the layer of a turbulent fluid most affected by interaction with a solid surface or the surface separating a gas and a liquid where the characteristics of the turbulence depend on distance from the interface. Surface layers are characterized by large normal gradients of tangential velocity and large concentration gradients of any substances transported to or from the interface.

<span class="mw-page-title-main">Planetary boundary layer</span> Lowest part of the atmosphere directly influenced by contact with the planetary surface

In meteorology, the planetary boundary layer (PBL), also known as the atmospheric boundary layer (ABL) or peplosphere, is the lowest part of the atmosphere and its behaviour is directly influenced by its contact with a planetary surface. On Earth it usually responds to changes in surface radiative forcing in an hour or less. In this layer physical quantities such as flow velocity, temperature, and moisture display rapid fluctuations (turbulence) and vertical mixing is strong. Above the PBL is the "free atmosphere", where the wind is approximately geostrophic, while within the PBL the wind is affected by surface drag and turns across the isobars.

<span class="mw-page-title-main">Large eddy simulation</span> Mathematical model for turbulence

Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer.

<span class="mw-page-title-main">Turbulence modeling</span> Use of mathematical models to simulate turbulent flow

In fluid dynamics, turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real-life scenarios. In spite of decades of research, there is no analytical theory to predict the evolution of these turbulent flows. The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real-life turbulent flows, CFD simulations use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations. In the Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a turbulence model.

The Obukhov length is used to describe the effects of buoyancy on turbulent flows, particularly in the lower tenth of the atmospheric boundary layer. It was first defined by Alexander Obukhov in 1946. It is also known as the Monin–Obukhov length because of its important role in the similarity theory developed by Monin and Obukhov. A simple definition of the Monin-Obukhov length is that height at which turbulence is generated more by buoyancy than by wind shear.

<span class="mw-page-title-main">Eddy diffusion</span> Mixing of fluids due to eddy currents

In fluid dynamics, eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which fluid substances mix together due to eddy motion. These eddies can vary widely in size, from subtropical ocean gyres down to the small Kolmogorov microscales, and occur as a result of turbulence. The theory of eddy diffusion was first developed by Sir Geoffrey Ingram Taylor.

The turbulent Prandtl number (Prt) is a non-dimensional term defined as the ratio between the momentum eddy diffusivity and the heat transfer eddy diffusivity. It is useful for solving the heat transfer problem of turbulent boundary layer flows. The simplest model for Prt is the Reynolds analogy, which yields a turbulent Prandtl number of 1. From experimental data, Prt has an average value of 0.85, but ranges from 0.7 to 0.9 depending on the Prandtl number of the fluid in question.

Ocean dynamics define and describe the flow of water within the oceans. Ocean temperature and motion fields can be separated into three distinct layers: mixed (surface) layer, upper ocean, and deep ocean.

<span class="mw-page-title-main">Mixing length model</span> Mathematical model in fluid dynamics

In fluid dynamics, the mixing length model is a method attempting to describe momentum transfer by turbulence Reynolds stresses within a Newtonian fluid boundary layer by means of an eddy viscosity. The model was developed by Ludwig Prandtl in the early 20th century. Prandtl himself had reservations about the model, describing it as, "only a rough approximation," but it has been used in numerous fields ever since, including atmospheric science, oceanography and stellar structure.

<span class="mw-page-title-main">Double diffusive convection</span> Convection with two density gradients

Double diffusive convection is a fluid dynamics phenomenon that describes a form of convection driven by two different density gradients, which have different rates of diffusion.

In fluid thermodynamics, combined forced convection and natural convection, or mixed convection, occurs when natural convection and forced convection mechanisms act together to transfer heat. This is also defined as situations where both pressure forces and buoyant forces interact. How much each form of convection contributes to the heat transfer is largely determined by the flow, temperature, geometry, and orientation. The nature of the fluid is also influential, since the Grashof number increases in a fluid as temperature increases, but is maximized at some point for a gas.

Representations of the atmospheric boundary layer in global climate models play a role in simulations of past, present, and future climates. Representing the atmospheric boundary layer (ABL) within global climate models (GCMs) are difficult due to differences in surface type, scale mismatch between physical processes affecting the ABL and scales at which GCMs are run, and difficulties in measuring different physical processes within the ABL. Various parameterization techniques described below attempt to address the difficulty in ABL representations within GCMs.

Monin–Obukhov (M–O) similarity theory describes the non-dimensionalized mean flow and mean temperature in the surface layer under non-neutral conditions as a function of the dimensionless height parameter, named after Russian scientists A. S. Monin and A. M. Obukhov. Similarity theory is an empirical method that describes universal relationships between non-dimensionalized variables of fluids based on the Buckingham π theorem. Similarity theory is extensively used in boundary layer meteorology since relations in turbulent processes are not always resolvable from first principles.

Open ocean convection is a process in which the mesoscale ocean circulation and large, strong winds mix layers of water at different depths. Fresher water lying over the saltier or warmer over the colder leads to the stratification of water, or its separation into layers. Strong winds cause evaporation, so the ocean surface cools, weakening the stratification. As a result, the surface waters are overturned and sink while the "warmer" waters rise to the surface, starting the process of convection. This process has a crucial role in the formation of both bottom and intermediate water and in the large-scale thermohaline circulation, which largely determines global climate. It is also an important phenomena that controls the intensity of the Atlantic Meridional Overturning Circulation (AMOC).

References

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